Equivalent magnetic network modeling solving method for planar induction motor

Abstract

The invention relates to an equivalent magnetic network modeling solving method for a planar induction motor. The method comprises the following steps: dividing an electromagnetic field solving area of the planar induction motor; giving size structure parameters and input parameters of the planar induction motor, and establishing an axial magnetic flux three-dimensional cylindrical coordinate system; establishing a preliminary magnetic network model, determining position angle information of grid nodes, and performing numerical solution of a cross-shaped magnetic conductance unit; according to the air gap layer connection node position angle information, adaptively connecting an air gap dynamic node; carrying out cubic spline interpolation solution on the magnetic conductivity of the nonlinear magnetic conductance unit based on a nonlinear ferromagnetic material B-H curve, and carrying out iterative solution through a relaxation coefficient method; and after iteration is completed, air gap flux density solving is realized.

Description

Technical Field

[0001] The invention belongs to the technical field of motor modeling and electromagnetic field solution, and specifically relates to an equivalent magnetic network modeling and solution method for a planar induction motor. Background Art

[0002] With the development of deep space exploration missions, large deployable mechanisms are more widely used in spacecraft. In order to meet their stress-free assembly and performance testing, they may tilt due to environmental factors during ground microgravity simulation tests. With the continuous development of motor technology, the working scenarios of motors are also constantly changing. Planar induction motors can generate horizontal traveling magnetic fields and can generate planar electromagnetic forces to compensate for the gravity component during tilt.

[0003] With the continuous development of computer technology, numerical methods represented by the finite element method have been widely used in the initial design stage of motors and in the field of electromagnetic field solution, but they also have the problems of long calculation time and large memory consumption. In contrast, the equivalent magnetic network method can better balance the requirements in terms of electromagnetic performance solution accuracy and calculation time. In the process of traditional equivalent magnetic network modeling and solution, the modeling and solution of the air gap part is a key research direction. The earliest modeling of the air gap part magnetic permeance was by single radial magnetic permeance connection, but this method has the disadvantage of not being able to simulate the tangential component, and there will be a sudden change in the magnetic permeance connection, which affects the solution of the electromagnetic field. In order to solve the deflection problem of magnetic lines of force in the air gap area, the concept of cross magnetic permeance is proposed. This modeling method can solve the magnetic field in two directions, but the cross magnetic permeance is still connected to the upper node by direct connection, and there is a sudden change in the magnetic permeance connection state. In the context of dynamic changes in magnetic network nodes, in view of the sudden change in magnetic permeance connection in the traditional method, the connection between the air gap node and the core node is mostly connected by a single-direction magnetic permeance unit using the Ostovic empirical coefficient. Although the unidirectional magnetic conductivity connection has the advantage of easy solution, it cannot deal with the problem of magnetic field line deflection in the connection layer.

[0004] In summary, it is particularly important to propose an accurate and effective equivalent magnetic network method that can achieve more accurate air gap magnetic field solution and motor performance prediction in the motor design research process. Summary of the invention

[0005] The present invention discloses a method for modeling and solving an equivalent magnetic network of a planar induction motor, which can accurately solve the air gap magnetic field of the planar induction motor and predict the motor performance, and effectively shorten the calculation time in the motor analysis and design stage. The technical solution is as follows:

[0006] A method for modeling and solving an equivalent magnetic network for a planar induction motor, characterized in that it comprises the following steps:

[0007] Step 1: Divide the electromagnetic field solution area of ​​the planar induction motor;

[0008] Step 2: Given the size, structure and input parameters of the planar induction motor, establish a three-dimensional cylindrical coordinate system for the axial flux;

[0009] Step 3: Establish a preliminary magnetic network model, determine the grid node position angle information, and perform numerical solution of the cross-permeability unit;

[0010] Step 4: Adaptively connect the air gap dynamic nodes according to the position angle information of the air gap layer connection nodes. The method is as follows:

[0011] According to the position angle information of the nodes connected to the preliminary magnetic network model and the air gap area, the position information between nodes is divided into overlap, intersection and separation based on the angle difference of the axis of each magnetic permeability unit node. According to the position relationship, the dynamic node layer of the air gap is obtained, and the adjacent node layers are adaptively connected to complete the construction of the equivalent magnetic network model.

[0012] When the node angle between the primary side and the air gap layer changes, the number of nodes and the value of the magnetic permeability unit change dynamically, and the angle difference of the axis of the magnetic permeability unit node is expressed as follows:

[0013] γ pg =θ p -θ g

[0014] In the formula, θ p is the primary side position angle, θ g is the air gap position angle;

[0015] The magnetic permeability unit is calculated according to the air gap adaptive mode as follows:

[0016]

[0017] In the formula, G zpg is the normal permeability unit in the air gap adaptive mode, G tpg is the tangential permeability unit in the air gap adaptive mode, θ pg It is expressed as:

[0018]

[0019] Step 5: Based on the BH curve of nonlinear ferromagnetic materials, the permeability of the nonlinear permeability unit is solved by cubic spline interpolation, and the relaxation coefficient method is used for iterative solution;

[0020] Step 6: After the iteration is completed, the air gap magnetic flux density is solved

[0021] After the convergence conditions are met, the calculation results at the current position are stored and the next position iteration is entered until all positions are solved at all times to achieve the air gap magnetic flux solution.

[0022] Furthermore, in step 1, the divided planar induction motor electromagnetic field solution area includes: a yoke, a tooth, a slot, a tooth top, a pole shoe, a notch, an air gap and a cast iron plate.

[0023] Furthermore, in step 2, the established axial flux three-dimensional cylindrical coordinate system is: the origin is the center point of the planar induction motor ring structure, and the coordinates of the coordinate system are (θ, r, z), where r is the horizontal distance from the moving point to the cylindrical axis, θ is the azimuth from the moving point to the reference direction in the same radius r direction on the horizontal plane, and z is the vertical distance from the moving point to the horizontal plane where the origin is located.

[0024] Furthermore, the method of step 3 is as follows:

[0025] 1) Calculation of the magnetic potential of the tooth

[0026] Given current excitation, calculate the slot current value and tooth magnetic potential in turn according to the power-on method:

[0027]

[0028] Where, Q is the number of primary side slots, N is a is the number of coil turns on a single tooth, F sj is the equivalent magnetomotive force source on the jth tooth, I sj is the equivalent current in the jth slot, i.e., the sum of the positive and negative currents in the slot;

[0029] 2) Solving the magnetic permeability unit

[0030] According to the direction of magnetic flux and the characteristics of the material flowing through, the magnetic field is analyzed by ring division, and the solution area is expanded along the radial direction;

[0031] According to whether the solution area is a linear area, the property of the magnetic permeability unit is set to be nonlinear or linear on each annular surface. Considering the distribution of magnetic lines of force in the planar induction motor, a cross magnetic permeability unit including tangential and normal magnetic permeabilities is selected for solution. The normal magnetic permeability G z and tangential permeability G t The expression is as follows:

[0032]

[0033] Wherein, μ is the magnetic permeability, θ is the angle corresponding to the magnetic permeability unit, R is the radius corresponding to the magnetic permeability unit in each ring section, L is the length of the magnetic permeability unit in the radial direction, and h is the height of the magnetic permeability unit;

[0034] 3) Calculation of magnetic flux source:

[0035] φ=G·F

[0036] Ф is the magnetic flux source placed on the tooth, G is the tooth magnetic permeance, and F is the tooth magnetic potential;

[0037] 4) Connect the magnetic network model:

[0038] The nonlinear permeability unit, linear permeability unit, and magnetic flux source are connected to complete the construction of the preliminary magnetic network model. The grid node position angle information is determined according to the number of permeability units in each area and the corresponding angle.

[0039] Furthermore, the ring-by-ring magnetic field analysis method is based on the following assumptions:

[0040] a) The current excitation given by the motor is evenly distributed in the slot. It is assumed that there are only axial flux and circumferential flux in the motor magnetic circuit, and the radial flux is ignored;

[0041] b) The magnetic field only exists in the annular analysis area, and the end effect of the magnetic field is ignored;

[0042] c) When dynamic alternating current is passed, eddy currents generated in ferromagnetic materials are ignored.

[0043] Furthermore, the method of step 5 is:

[0044] 1) Based on the node current method, the magnetic permeance, magnetic potential and magnetic flux matrix equations are established:

[0045]

[0046] In the formula, in the sub-permeability matrix G ii In the matrix, the diagonal elements represent the self-permeability of each node in the i-th layer, and the off-diagonal elements represent the negative value of the mutual permeability between the nodes in the i-th layer; the sub-permeability matrix G ij represents the negative value of the mutual magnetic permeability between the nodes in the i-th layer and the nodes in the j-th layer, and the sub-column vector F i T is the magnetic potential of each node in the i-th layer, and the sub-column vector Φ i T is the equivalent magnetic flux source of each node in the i-th layer;

[0047] 2) Solve the equation to obtain the magnetic potential of each node, calculate the magnetic potential difference of the nonlinear permeability unit based on the node magnetic potential, and solve to obtain the unit magnetic field strength;

[0048] 3) Based on the BH curve of the given ferromagnetic material, the cubic spline interpolation method is used to solve the unit magnetic induction intensity of the nonlinear class, and further obtain the unit magnetic permeability;

[0049] 4) The relaxation coefficient method is used to iterate the magnetic permeability until the error requirement is met. The expression is as follows:

[0050] μ k+1 =αμ k +(1-α)μk-1

[0051] In the formula, μ k+1 is the material magnetic permeability obtained by relaxation iteration, k is the current iteration number, α is the relaxation coefficient, 0<α<1.

[0052] Beneficial effects of the present invention:

[0053] First, the equivalent magnetic network modeling and solution method of the planar induction motor of the present invention is based on the magnetic circuit analysis of the motor. Compared with the finite element method, the computing resource usage and solution time are greatly reduced, and the analysis efficiency is effectively improved in the motor design optimization stage.

[0054] Second, compared with the existing equivalent magnetic network modeling that uses unidirectional magnetic permeance in the air gap and iron core connection layer, the equivalent magnetic network modeling and solution method of the planar induction motor of the present invention takes into account the deflection of magnetic lines of force in the air gap and iron core connection layer. When establishing the network model, an adaptive cross-connection magnetic permeance is used, which can effectively solve the axial and tangential components of the air gap magnetic density of the connection layer and achieve better solution accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 is a schematic diagram of a planar induction motor model;

[0056] Figure 2 It is the distribution diagram of the solution area of ​​the planar induction motor;

[0057] Figure 3 It is the radial ring cross-section development;

[0058] Figure 4 It is a diagram of the magnetic network model of a planar induction motor;

[0059] Figure 5 is a schematic diagram of a magnetic permeability unit;

[0060] Figure 6 It is a schematic diagram of the nonlinear change of the adaptive cross-conductance and the node connection process;

[0061] Figure 7 It is a schematic diagram of nodes of magnetic network model;

[0062] Figure 8 It is a schematic diagram of magnetic network iteration;

[0063] Fig. 9 It is a comparison chart between the air gap magnetic flux calculated by the present invention and the finite element method analysis result. DETAILED DESCRIPTION

[0064] The following examples will enable those skilled in the art to more fully understand the present invention, but are not intended to limit the present invention in any way.

[0065] The present invention provides an equivalent magnetic network modeling and solving method for a planar induction motor, comprising two parts: equivalent magnetic network modeling based on an equivalent magnetic circuit method of an adaptive air gap connection layer, and iterative solving. The equivalent magnetic network model of the planar induction motor includes four parts: the magnetic permeability of the primary side core part, the magnetic permeability of the secondary side cast iron plate, the air magnetic permeability of the slot area, and the air gap magnetic permeability. After the equivalent magnetic network model is established, the magnetic permeability of the ferromagnetic material is solved by cubic spline interpolation according to the BH curve of the ferromagnetic material, and the magnetic permeability of the nonlinear ferromagnetic material area is iteratively solved by a relaxation iteration method. In this way, the magnetic field analysis of the planar induction motor based on the equivalent magnetic network is realized, and the calculation results are well consistent with the finite element results.

[0066] Figure 1 This is a topological diagram of the planar induction motor of the embodiment of the present invention. The motor system consists of an air bearing housing, porous graphite, an iron core, a winding, and a magnetic and conductive cast iron plate. The iron core is a semi-closed parallel slot structure, including a yoke, a tooth, a slot and other regions. The motor is a 12-slot motor, and the winding is wound around the tooth in the form of a concentrated winding. The parameters of the planar induction motor in the embodiment of the present invention are shown in Table 1.

[0067] Table 1 Parameters of the planar induction motor in the embodiment of the present invention

[0068]

[0069]

[0070] exist Figure 1 Based on the topological structure diagram of the example of the present invention shown, the present invention divides the motor structure according to the motor topological structure and the material characteristics of each part.

[0071] According to Figure 1 The topological structure of the example of the present invention is shown. In the process of analyzing the motor, the present invention adopts the method of split ring analysis. The radial ring cross-section expansion diagram is shown in FIG. Figure 2 shown.

[0072] exist Figure 2 In the context of the circular analysis shown, Figure 3 As shown, the electromagnetic field solution area is divided into 9 parts: primary side yoke, primary side tooth part, primary side slot part, primary side tooth top, primary side pole shoe, primary side slot, air gap, and cast iron plate.

[0073] Based on the magnetic field distribution characteristics of the planar induction motor, an axial magnetic flux three-dimensional cylindrical coordinate system is established. The origin is the center point of the annular structure of the planar induction motor. The coordinates of the coordinate system are (θ, r, z), where r is the horizontal distance from the moving point to the cylindrical axis, θ is the azimuth from the moving point to the reference direction in the horizontal plane with the same radius r, and z is the vertical distance from the moving point to the horizontal plane where the origin is located. The size and structural parameters of the planar induction motor are given.

[0074] According to the direction of magnetic flux and the characteristics of the material flowing through it, and considering the density of magnetic flux distribution in different regions, nonlinear magnetic permeability and linear magnetic permeability units are set respectively and the number of magnetic permeability units is given;

[0075] The flux source can be written as:

[0076] φ=G·F

[0077] In the formula, Ф is the magnetic flux source placed on the tooth, G is the tooth magnetic permeance, and F is the tooth magnetic potential.

[0078] Given current excitation, calculate the slot current value and tooth magnetic potential in turn according to the power-on method:

[0079]

[0080] Where, Q is the number of primary side slots, N is a is the number of coil turns on a single tooth, F sj is the equivalent magnetomotive force source on the jth tooth, I sj is the equivalent current in the jth slot, that is, the sum of the positive and negative currents in the slot.

[0081] Figure 4 This is a diagram of the magnetic network model of a planar induction motor. As shown in the figure, the nonlinear magnetic permeability unit, the linear magnetic permeability unit, and the magnetic flux source are connected to preliminarily establish a magnetic network model. According to the number of magnetic permeability units in each area, the grid node position information is determined to perform numerical solutions for the magnetic permeability units.

[0082] Figure 5 is a schematic diagram of the magnetic permeability unit. The expression is as follows:

[0083]

[0084] Where μ is the magnetic permeability, θ is the angle corresponding to the magnetic permeability unit, R is the radius corresponding to the magnetic permeability unit, L is the length of the magnetic unit in the radial direction, and h is the height of the magnetic unit.

[0085] According to the position angle information of the nodes connected to the magnetic network model and the air gap area, the angle difference of the axis of each node is used as the basis for judgment, and the following is obtained: Figure 6 The position information between nodes shown is divided into overlap, intersection and separation, and the air gap dynamic node layer is obtained according to the position relationship.

[0086] The air gap dynamic node layer is based on the change of the node angle between the primary side and the air gap layer. The number of nodes and the value of the magnetic permeability unit are dynamically transformed. The node angle information difference can be expressed as follows:

[0087] γ pg =θ p -θ g

[0088] In the formula, θ p is the primary side node position angle, θ g is the air gap node position angle.

[0089] Under the background of power on, the magnetic lines of force will be deflected at the connection between the nonlinear ferromagnetic material and the air gap, so the adaptive air gap permeance adopts the form of cross permeance, and the calculation is as follows:

[0090]

[0091] In the formula, θpg can be expressed as:

[0092]

[0093] Adaptively connect adjacent node layers to complete the construction of the magnetic network model. The schematic diagram of the magnetic network model node is as follows Figure 7 shown.

[0094] Thus, the equivalent magnetic network model of the planar induction motor of the entire embodiment is established. Based on the node current method, the magnetic permeance, magnetic potential, and magnetic flux matrix equations are established:

[0095]

[0096] In the formula, in the sub-permeability matrix G ii In the matrix G, the diagonal elements represent the self-permeability of each node in the i-th layer, and the off-diagonal elements represent the negative value of the mutual permeability between the nodes in the i-th layer. ij represents the negative value of the mutual magnetic permeability between the nodes in the i-th layer and the nodes in the j-th layer, and the sub-column vector F i T is the magnetic potential of each node in the i-th layer, and the sub-column vector Φ i T is the equivalent magnetic flux source of each node in the i-th layer.

[0097] Solve the equation to obtain the magnetic potential of each node, calculate the magnetic potential difference of the magnetic permeability unit according to the node magnetic potential, and solve to obtain the unit magnetic field strength;

[0098] Figure 8 Schematic diagram of magnetic network iteration. The steps of magnetic network iteration in the embodiment are as follows:

[0099] Step 1: Parameterize the planar induction motor used in the example of the present invention.

[0100] Step 2: According to the topological structure of the example of the present invention, iterations are performed at time t and radius r respectively during the iteration process using the time-stepping method.

[0101] Step 3: Determine the positional relationship between nodes based on the grid node position information.

[0102] Step 4: Adaptively connect the air gap nodes according to the positional relationship between the nodes.

[0103] Step 5: Solve the magnetic network equation to obtain the magnetic field strength H of each node unit.

[0104] Step 6: According to the nonlinear ferromagnetic material BH curve, use cubic spline to interpolate and solve the node unit magnetic induction intensity B and magnetic permeability μ k .

[0105] Step 7: Use the relaxation coefficient method to iterate the magnetic permeability until the error requirement is met. The expression is as follows:

[0106] μ k+1 =αμ k +(1-α)μ k-1

[0107] In the formula, μ k+1 is the material magnetic permeability obtained by relaxation iteration, k is the current iteration number, α is the relaxation coefficient, 0<α<1.

[0108] Step 8: After the convergence conditions are met, store the calculation results at the current position and enter the next position iteration until all positions are solved at all times, store the node values, and realize the solution of the air gap magnetic flux density.

[0109] Fig. 9 This is a comparison chart of the air gap magnetic flux calculated by the present invention and the finite element method analysis results. The air gap magnetic flux waveforms obtained by the two methods are highly consistent, indicating that the equivalent magnetic network solution method based on adaptive connection of air gap dynamic nodes has high calculation accuracy and can be expected to become an effective alternative to the finite element method in the analysis process.

[0110] The present invention is based on an equivalent magnetic network solution method. In view of the problem that the traditional equivalent magnetic network using a single radial flux connection air gap layer obtained by an empirical formula cannot accurately reflect the deflection of magnetic lines of force, a magnetic network modeling idea using dynamic connection of cross flux units is proposed, and the electromagnetic performance of the planar induction motor represented by the axial component of the air gap flux density is solved and calculated.

[0111] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.

Claims

1. A method for modeling and solving an equivalent magnetic network for a planar induction motor, characterized in that: The following steps are involved: Step 1: Divide the electromagnetic field solution area of ​​the planar induction motor; Step 2: Given the size, structure and input parameters of the planar induction motor, establish a three-dimensional cylindrical coordinate system for the axial flux; Step 3: Establish a preliminary magnetic network model, determine the grid node position angle information, and perform numerical solution of the cross-permeability unit; Step 4: Adaptively connect the air gap dynamic nodes according to the position angle information of the air gap layer connection nodes. The method is as follows: According to the position angle information of the nodes connected to the preliminary magnetic network model and the air gap area, the position information between nodes is divided into overlap, intersection and separation based on the angle difference of the axis of each magnetic permeability unit node. According to the position relationship, the dynamic node layer of the air gap is obtained, and the adjacent node layers are adaptively connected to complete the construction of the equivalent magnetic network model. When the node angle between the primary side and the air gap layer changes, the number of nodes and the value of the magnetic permeability unit change dynamically, and the angle difference of the axis of the magnetic permeability unit node is expressed as follows: c pg =θ p -θ g In the formula, θ p is the primary side position angle, θ g is the air gap position angle; The magnetic permeability unit is calculated according to the air gap adaptive mode as follows: In the formula, G zpg is the normal permeability unit in the air gap adaptive mode, G tpg is the tangential permeability unit in the air gap adaptive mode, θ pg It is expressed as: Step 5: Based on the BH curve of nonlinear ferromagnetic materials, the permeability of the nonlinear permeability unit is solved by cubic spline interpolation, and the relaxation coefficient method is used for iterative solution; Step 6: After the iteration is completed, the air gap magnetic flux density is solved After the convergence conditions are met, the calculation results at the current position are stored and the next position iteration is entered until all positions are solved at all times to achieve the air gap magnetic flux solution.

2. The method for modeling and solving the equivalent magnetic network of a planar induction motor according to claim 1, characterized in that: In step 1, the divided planar induction motor electromagnetic field solution area includes: yoke, tooth, slot, tooth top, pole shoe, slot, air gap and cast iron plate.

3. The equivalent magnetic network modeling and solving method of the planar induction motor according to claim 1 is characterized in that: In step 2, the established axial flux three-dimensional cylindrical coordinate system is as follows: the origin is the center point of the planar induction motor ring structure, and the coordinates of the coordinate system are (θ, r, z), where r is the horizontal distance from the moving point to the cylindrical axis, θ is the azimuth from the moving point to the reference direction in the same radius r direction on the horizontal plane, and z is the vertical distance from the moving point to the horizontal plane where the origin is located.

4. The method for modeling and solving the equivalent magnetic network of a planar induction motor according to claim 1, characterized in that: The method for step 3 is as follows: 1) Calculation of the magnetic potential of the tooth Given current excitation, calculate the slot current value and tooth magnetic potential in turn according to the power-on method: Where, Q is the number of primary side slots, N is a is the number of coil turns on a single tooth, F sj is the equivalent magnetomotive force source on the jth tooth, I sj is the equivalent current in the jth slot, i.e., the sum of the positive and negative currents in the slot; 2) Solving the magnetic permeability unit According to the direction of magnetic flux and the characteristics of the material flowing through, the magnetic field is analyzed by ring division, and the solution area is expanded along the radial direction; According to whether the solution area is a linear area, the property of the magnetic permeability unit is set to be nonlinear or linear on each annular surface. Considering the distribution of magnetic lines of force in the planar induction motor, a cross magnetic permeability unit including tangential and normal magnetic permeabilities is selected for solution. The normal magnetic permeability G z and tangential permeability G t The expression is as follows: Wherein, μ is the magnetic permeability, θ is the angle corresponding to the magnetic permeability unit, R is the radius corresponding to the magnetic permeability unit in each ring section, L is the length of the magnetic permeability unit in the radial direction, and h is the height of the magnetic permeability unit; 3) Calculation of magnetic flux source: φ=G·F Ф is the magnetic flux source placed on the tooth, G is the tooth magnetic permeance, and F is the tooth magnetic potential; 4) Connect the magnetic network model: The nonlinear permeability unit, linear permeability unit, and magnetic flux source are connected to complete the construction of the preliminary magnetic network model. The grid node position angle information is determined according to the number of permeability units in each area and the corresponding angle.

5. The method for modeling and solving the equivalent magnetic network of a planar induction motor according to claim 4, characterized in that: The above-mentioned ring-by-ring magnetic field analysis method is based on the following assumptions: a) The current excitation given by the motor is evenly distributed in the slot. It is assumed that there are only axial flux and circumferential flux in the motor magnetic circuit, and the radial flux is ignored; b) The magnetic field only exists in the annular analysis area, and the end effect of the magnetic field is ignored; c) When dynamic alternating current is passed, eddy currents generated in ferromagnetic materials are ignored.

6. The method for modeling and solving the equivalent magnetic network of a planar induction motor according to claim 4, characterized in that: The method for step 5 is: 1) Based on the node current method, the magnetic permeance, magnetic potential and magnetic flux matrix equations are established: In the formula, in the sub-permeability matrix G ii In the matrix, the diagonal elements represent the self-permeability of each node in the i-th layer, and the off-diagonal elements represent the negative value of the mutual permeability between the nodes in the i-th layer; the sub-permeability matrix G ij represents the negative value of the mutual magnetic permeability between the nodes in the i-th layer and the nodes in the j-th layer, and the sub-column vector F i T is the magnetic potential of each node in the i-th layer, and the sub-column vector Φ i T is the equivalent magnetic flux source of each node in the i-th layer; 2) Solve the equation to obtain the magnetic potential of each node, calculate the magnetic potential difference of the nonlinear permeability unit based on the node magnetic potential, and solve to obtain the unit magnetic field strength; 3) Based on the BH curve of the given ferromagnetic material, the cubic spline interpolation method is used to solve the unit magnetic induction intensity of the nonlinear class, and further obtain the unit magnetic permeability; 4) The relaxation coefficient method is used to iterate the magnetic permeability until the error requirement is met. The expression is as follows: m k+1 =am k +(1-a)m k-1 In the formula, μ k+1 is the material magnetic permeability obtained by relaxation iteration, k is the current iteration number, α is the relaxation coefficient, 0<α<1.

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